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Full text of "Analytical Mechanics"

PERIODIC MOTION                           329
mass.   Find the period with which the particle will vibrate when displaced along the string.
Let L' be the stretched length of the string, A the area of its cross-section, q the distance of the particle from its position of equilibrium, and T\ and jP2 the tensile forces of the two parts of the string. Then by Hooke's law we have
£'      \    L
1     y   I         n              T t         7"       !      O   ~
T = X -----r-A               L
2 __ , L' -L~-2q \
2 Therefore the resultant force on the particle is
T: A
=      -i^ -- ITg==~~L" where X' = ^LX.   Hence the potential energy equals
But since the kinetic energy is given by
1           2X'
we obtain                       E = - mq* + — q2
Al                                 JLJ
for the total energy of the system.   Differentiating the last equation we get                                     mg + ^-o
which gives                                       ___
q^asm^—tt + to)
and                                P =
3. A cylinder performs small oscillations inside of a fixed cylinder. Find the period of the motion, supposing the contact between the cylinders to be rough enough to prevent sliding.